{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T21:25:06Z","timestamp":1775856306330,"version":"3.50.1"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":13160,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1978,3]]},"abstract":"<jats:p>In this paper, we give consistency proofs for the existence of a \u03ba-saturated ideal on an inaccessible \u03ba, and for the existence of an \u03c9<jats:sub>2<\/jats:sub>-saturated ideal on \u03c9<jats:sub>1<\/jats:sub>. We also include an historical survey outlining other known results on saturated ideals.<\/jats:p><jats:p><jats:bold>\u00a71. Forcing<\/jats:bold>. We assume that the reader is familiar with the usual techniques in forcing and Boolean-valued models (see Jech [3] or Rosser [11]), so we shall just specify here some of the less standard notation.<\/jats:p><jats:p>\u201ccBa\u201d abbreviates \u201ccomplete Boolean algebra\u201d.<\/jats:p><jats:p>If <jats:italic>P<\/jats:italic> (= \u2039<jats:italic>P<\/jats:italic>, &lt;\u203a) is a notion of forcing, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049896_inline1\"\/> is the associated cBa. We write <jats:italic>V<jats:sup>P<\/jats:sup><\/jats:italic> for <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049896_inline2\"\/>.<\/jats:p><jats:p>Notions like \u03ba-closed, \u03ba-complete, etc. always mean &lt; \u03ba. Thus, <jats:italic>P<\/jats:italic> is \u03ba-closed iff every decreasing chain of length <jats:italic>less than<\/jats:italic> \u03ba has a lower bound, and a Boolean algebra <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049896_inline3\"\/> is \u03ba-complete iff sups of subsets of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049896_inline3\"\/> of cardinality <jats:italic>less than<\/jats:italic> \u03ba always exist.<\/jats:p><jats:p>If <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049896_inline3\"\/> is a cBa, x\u030c is the object in <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200049896_inline4\"\/> representing <jats:italic>x<\/jats:italic> in <jats:italic>V<\/jats:italic>. In many cases, especially with ordinals, the \u030c is dropped. V\u030c is the Boolean-valued class representing <jats:italic>V<\/jats:italic> \u2014 i.e.,<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200049896_eqnU1\"\/><\/jats:disp-formula><\/jats:p>","DOI":"10.2307\/2271949","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:46:34Z","timestamp":1146951994000},"page":"65-76","source":"Crossref","is-referenced-by-count":105,"title":["Saturated ideals"],"prefix":"10.1017","volume":"43","author":[{"given":"Kenneth","family":"Kunen","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200049896_ref007","first-page":"407","volume":"36","author":"Kunen","year":"1971","journal-title":"Elementary embeddings and infinitary combinations"},{"key":"S0022481200049896_ref002","first-page":"254","volume":"39","author":"Boos","year":"1974","journal-title":"Boolean extensions which efface the Mahlo property"},{"key":"S0022481200049896_ref001","unstructured":"Baumgartner J. E. , Results and independence proofs in combinatorial set theory, Doctoral Dissertation, University of California, Berkeley, 1970."},{"key":"S0022481200049896_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90013-6"},{"key":"S0022481200049896_ref014","doi-asserted-by":"publisher","DOI":"10.2307\/1970860"},{"key":"S0022481200049896_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(72)90001-0"},{"key":"S0022481200049896_ref011","volume-title":"Pure and Applied Mathematics","volume":"31","author":"Rosser","year":"1969"},{"key":"S0022481200049896_ref008","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(71)90001-5"},{"key":"S0022481200049896_ref009","unstructured":"Menas T. K. , On strong compactness and supercompactness, Doctoral Dissertation, University of California, Berkeley, 1973."},{"key":"S0022481200049896_ref004","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1971.39.395"},{"key":"S0022481200049896_ref003","volume-title":"Lecture Notes in Mathematics","author":"Jech","year":"1971"},{"key":"S0022481200049896_ref016","doi-asserted-by":"publisher","DOI":"10.4064\/fm-16-1-140-150"},{"key":"S0022481200049896_ref010","article-title":"Changing measurable into accessible cardinals","volume":"68","author":"Prikry","year":"1970","journal-title":"Dissertationes Mathematicae, Rozprawy Matematyczne"},{"key":"S0022481200049896_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1957-0084466-4"},{"key":"S0022481200049896_ref013","first-page":"397","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"13","author":"Solovay","year":"1967"},{"key":"S0022481200049896_ref015","doi-asserted-by":"publisher","DOI":"10.4064\/fm-33-1-51-65"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200049896","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T20:19:18Z","timestamp":1558988358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200049896\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,3]]},"references-count":16,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1978,3]]}},"alternative-id":["S0022481200049896"],"URL":"https:\/\/doi.org\/10.2307\/2271949","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,3]]}}}